Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein (eBook)
410 Seiten
Elsevier Science (Verlag)
978-1-4832-8270-1 (ISBN)
Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein aims to remedy the deficiency in geometry so that the ideas of F. Klein obtain the place they merit in the literature of mathematics. This book discusses the axioms of betweenness, lattice of linear subspaces, generalization of the notion of space, and coplanar Desargues configurations. The central collineations of the plane, fundamental theorem of projective geometry, and lines perpendicular to a proper plane are also elaborated. This text likewise covers the axioms of motion, basic projective configurations, properties of triangles, and theorem of duality in projective space. Other topics include the point-coordinates in an affine space and consistency of the three geometries. This publication is beneficial to mathematicians and students learning geometry.
Front Cover 1
Foundation of Euclidean and Non-Euclidean Geometries According to F. Klein 4
Copyright Page 5
Table of Contents 6
PREFACE 10
CHAPTER 1. AXIOMS 12
§ 1. Axioms of incidence 12
§ 2. Axioms of betweenness 13
§ 3. Axiom of continuity 14
§ 4. Axioms of motion 14
CHAPTER 2. CONSEQUENCES OF THE SYSTEM OF AXIOMS I 16
§ 5. Simple properties of straight lines and planes 16
§ 6. Desargues configurations 19
§ 7. Linear subspaces 22
§ 8. The lattice of linear subspaces 23
§ 9. Basic projective configurations 24
§ 10. Projection and intersection 26
CHAPTER 3. SIMPLE CONSEQUENCES OF THE SYSTEMS OF AXIOMS I, II 28
§ 11. Segments. Triangles 28
§ 12. Properties of segments 31
§ 13. Linear ordering 34
§ 14. Properties of triangles 40
§ 15. The tetrahedron 44
§ 16. Neighbourhoods 47
§ 17. Validity of the systems of Axioms I, II for the basic domain R' 48
§ 18. Generalization of the notion of space 50
§ 19. The extension and restriction of spaces 50
CHAPTER 4. PROJECTIVE CLOSURE 51
§ 20. Half-subspaces 51
§ 21. Half-pencils. Angles 56
§ 22. Some properties of pencils and bundles 61
§ 23. Coplanar Desargues configurations 62
§ 24. Improper pencils of lines 66
§ 25. Improper bundles of lines 72
§ 26. The projective closure R of R 76
§ 27. The projective axioms 95
§ 28. The general case 104
CHAPTER 5. INVESTIGATION OF THE PROJECTIVE SPACE 107
§ 29. Preliminaries 107
§ 30. Theorem of duality in projective space 109
§ 31. Collineations 110
§ 32. The Erlangen programme 113
§ 33. Theorem of duality of the plane 114
§ 34. Perspectivities and projectivities 116
§ 35. Central collineations of the plane 121
§ 36. Separation 133
§ 37. Cyclic ordering 144
§ 38. Projective segments and angles 149
§ 39. Complete quadrangles. Harmonic points 155
§ 40. Preliminaries about coordinate systems 164
§ 41. Coordinates in projective scales 174
§ 42. Halving a projective scale 179
§ 43. Coordinates for dyadic sets of points on a line 181
CHAPTER 6. CONSEQUENCES OF THE SYSTEMS OF AXIOMS I, II, III 186
§ 44. Preliminaries 186
§ 45. Theorem concerning the infinite point 191
§ 46. Coordinates in an affine line 196
§ 47. Coordinates on the basic projective configurations of the first degree 202
§ 48. Point-coordinates in an affine plane 205
§ 49. The fundamental theorem of projective geometry 214
§ 50. Point-coordinates in an affine space 223
§ 51. Vectors 229
§ 52. Homogeneous point- and plane-coordinates in space. Point- and line-coordinates in a plane 231
§ 53. Determination of all collineations of the space 241
§ 54. Determination of the coordinate transformations of space 245
§ 55. Transformation of projective coordinates 250
§ 56. Cross ratio 252
§ 57. Imaginary points 258
§ 58. Fixed elements of projectivities 259
§ 59. Involutions 260
§ 60. Involutory collineations of a plane 263
CHAPTER 7. CONSEQUENCES OF THE SYSTEMS OF AXIOMS I, II, III, IV 265
§ 61. Extended motions 265
§ 62. The comparability of segments 270
§ 63. Reflections and rotations. Absolute polar plane 275
§ 64. Metric scales. Infinite and ultra-infinite points. Elliptic, parabolic and hyperbolic geometries 286
§ 65. Absolute involution of points on a proper line 294
§ 66. Midpoint and bisector 299
§ 67. The lines perpendicular to a proper plane 304
§ 68. Motions as products of reflections 312
§ 69. Polarities with respect to surfaces and curves of the second order 314
§ 70. The absolute configuration in the elliptic case 322
§ 71. The absolute configuration in the hyperbolic case 324
§ 72. Characterization of motions in the non-parabolic case 333
§ 73. The absolute configuration and characterization of motions in the parabolic case 336
§ 74. Formulae of motion of the three geometries 348
§ 75. The consistency of the three geometries 363
§ 76. Measuring of segments 377
§ 77. Measuring of angles 387
§ 78. Applications to trigonometry 393
Bibliography 402
Index 404
Erscheint lt. Verlag | 15.7.2014 |
---|---|
Sprache | englisch |
Themenwelt | Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika |
Technik | |
ISBN-10 | 1-4832-8270-8 / 1483282708 |
ISBN-13 | 978-1-4832-8270-1 / 9781483282701 |
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